Can you use fixed effects models for time-invariant, multilevel data?

[Ozan Mattock]

I would just like to ask whether fixed effects models can be used for time-invariant, multilevel data instead of panel data (as an alternative to multilevel modelling)? In this case dummy variables would be included for all level 2 units.

My understanding is that the term, 'fixed effects', relates specifically to the units of observation and the panel data structure, but I don't see why this would be an issue for the model mathematically.


Many thanks,

Ozan

[Carlo Lazzaro]

Ozan:
welcome to this forum.
The issue is that -mixed- is equivalent to -xtreg, mle- (an -re- estimator), as you can see from the following toy-example:

. use "https://www.stata-press.com/data/r18/nlswork.dta"
(National Longitudinal Survey of Young Women, 14-24 years old in 1968)

. mixed ln_wage i.race, vce(cluster idcode)||idcode:

Performing EM optimization ...

Performing gradient-based optimization: 
Iteration 0:  Log pseudolikelihood = -12817.312  
Iteration 1:  Log pseudolikelihood = -12817.312  

Computing standard errors ...

Mixed-effects regression                             Number of obs    = 28,534
Group variable: idcode                               Number of groups =  4,711
                                                     Obs per group:
                                                                  min =      1
                                                                  avg =    6.1
                                                                  max =     15
                                                     Wald chi2(2)     = 102.63
Log pseudolikelihood = -12817.312                    Prob > chi2      = 0.0000

                             (Std. err. adjusted for 4,711 clusters in idcode)
------------------------------------------------------------------------------
             |               Robust
     ln_wage | Coefficient  std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
        race |
      Black  |  -.1301537   .0131255    -9.92   0.000    -.1558792   -.1044283
      Other  |   .1006124   .0662975     1.52   0.129    -.0293284    .2305532
             |
       _cons |   1.692089   .0069705   242.75   0.000     1.678427    1.705751
------------------------------------------------------------------------------

------------------------------------------------------------------------------
                             |               Robust           
  Random-effects parameters  |   Estimate   std. err.     [95% conf. interval]
-----------------------------+------------------------------------------------
idcode: Identity             |
                  var(_cons) |   .1362594   .0037256      .1291495    .1437607
-----------------------------+------------------------------------------------
               var(Residual) |   .1031165   .0024214      .0984783    .1079732
------------------------------------------------------------------------------

. xtreg ln_wage i.race, vce(cluster idcode) mle

Fitting constant-only model:
Iteration 0:  Log likelihood = -12883.463
Iteration 1:  Log likelihood = -12869.087
Iteration 2:  Log likelihood = -12868.942
Iteration 3:  Log likelihood = -12868.942

Fitting full model:
Iteration 0:  Log likelihood = -12833.705
Iteration 1:  Log likelihood = -12817.493
Iteration 2:  Log likelihood = -12817.312
Iteration 3:  Log likelihood = -12817.312

Random-effects ML regression                         Number of obs    = 28,534
Group variable: idcode                               Number of groups =  4,711

Random effects u_i ~ Gaussian                        Obs per group:
                                                                  min =      1
                                                                  avg =    6.1
                                                                  max =     15

                                                     Wald chi2(2)     = 102.63
Log likelihood = -12817.312                          Prob > chi2      = 0.0000

                             (Std. err. adjusted for 4,711 clusters in idcode)
------------------------------------------------------------------------------
             |               Robust
     ln_wage | Coefficient  std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
        race |
      Black  |  -.1301537   .0131265    -9.92   0.000    -.1558812   -.1044263
      Other  |   .1006124   .0663095     1.52   0.129    -.0293519    .2305767
             |
       _cons |   1.692089   .0069623   243.04   0.000     1.678443    1.705735
-------------+----------------------------------------------------------------
    /sigma_u |   .3691333   .0050312                      .3594028    .3791272
    /sigma_e |   .3211176   .0037712                      .3138106    .3285947
         rho |   .5692276   .0091414                      .5512457    .5870672
------------------------------------------------------------------------------

.

Therefore, no wonder that coefficients of time-invariant predictors can be calculated.

[Jeff Wooldridge]

Fixed effects estimation is often applied to non-panel structures. For example, you have students nested within schools for one years. You can include school fixed effects to estimate coefficients on variables that change at the student level. And then cluster by school.